Some uniqueness results for degenerate elliptic operators in two variables (Q1192076)

The authors' aim is to provide unique continuation across \(t=0\) for degenerate second order operators of the form \[ P(x,t,D_ x,D_ t)=D_ t^ 2+a(t)D_ x^ 2+ [b_ 1(t)+ib_ 2(t)] D_ x+c(x,t)D_ t+d(x,t), \] where \(a\in C^ 1(\mathbb{R},[0,\infty))\); \(b_ i\in C^ 1(\mathbb{R},\mathbb{R})\); \(c,d\in L^ \infty(\Omega,\mathbb{C})\); \(\Omega\) open neighbourhood of the origin of \(\mathbb{R}^ 2\). The proof combines Carleman estimates with suitable assumptions concerning the degeneracy of the coefficients \(a\), \(b_ i\) at \(t=0\). These assumptions are too complicated to be stated here. A typical example of an operator \(P\) to which the authors' results can be applied (whereas earlier work cannot) is \[ P(x,t,D_ x,D_ t)=D_ t^ 2+\exp(-1/t^ 2)D_ x^ 2+\exp(- 1/t)(1+it^ 2)D_ x+\dots\;. \]